To the best of our knowledge, the concept of stable solutions has not been used yet in real KEPs. In fact, besides considering some logistical and quality factors, the typical objective in European programs is to maximize the number of transplants [1]}, [2]}. However, there are several reasons why stable solutions may well be aimed for in the future. One reason is that quality of the transplants is becoming more important for the patients and system organizers as well, since increasing the expected graft survival time yields a longer expected life time for the patient and reduces the risk that this patient will move back to the waiting list for another transplant, in a state when it becomes more complicated to find a compatible donor. Secondly, there are more and more platforms where the patient-donor pairs can communicate, and so the possibility of finding a better exchange for some pairs may become feasible through alternative channels.
Finally, there is an increasing number of compatible pairs or ABO-incompatible pairs, who are joining the KEP pools instead of direct transplantation with the incentive of getting a better donor. In the UK, for example, this strict improvement can be guaranteed by the possibility of setting quality thresholds, which is an available option for every pair. A pair might strategically set its threshold high enough in the first matching run it participates in and then they gradually decrease it in later rounds if no exchanges are offered. Indeed, the actual solution can be close to being stable in such a dynamic system, even if the main optimization goal is to maximize the number of transplants. However, as this process takes time to converge, the health condition of the patients gets worse while waiting for convergence. A direct mechanism for finding stable exchanges can give the incentive to the pairs of reporting low thresholds, as they are guaranteed to get the best possible donor in a stable solution.
In fact, if no length limit is imposed then the unique strongly stable solution, obtained by the TTC algorithm satisfies the respecting improvement property, which means that bringing a better quality (e.g. younger) or more compatible (e.g. blood-type O) donor can only improve the quality of the exchange donor that this patient will receive [3]}. This property is violated for the standard objectives of maximizing the size or the total weight (e.g., quality score) of the solution, which can be a serious concern.
